APSTA-GE 2003: Intermediate Quantitative Methods

class: center, middle, inverse, title-slide

# APSTA-GE 2003: Intermediate Quantitative Methods
## Lab Section 003, Week 2
### New York University
### 09/15/2020

---

## Reminders

- Assignment 1
  - Due: **09/23/2020 11:55pm (EST)**
- Assignment 2 is available now
  - Due: **10/02/2020 11:55pm (EST)**
- Office hours
  - Monday 9 - 10am (EST)
  - Wednesday 12:30 - 1:30pm (EST)
- Office hour notes
  - Available on NYU Classes under the "Resources" tab

---

## Fun slide

**"Get ten bagels when you arrive at the bakery. If you see muffins, get one."**

How many bagels will you buy?

- A. 10

- B. 1

---

## Another fun slide

**"Get ten bagels when you arrive at the bakery. If you see muffins, get one."**

```r
has_Muffin <- TRUE
shopping_at <- function (location) {
 if (location == "bakery") {
 if (has_Muffin == TRUE) {
 bagel <- 1
 } else {
 bagel <- 10
 }
 }
 return(bagel)
}
shopping_at(location = "bakery")
```

```
## [1] 1
```

---

## One more fun slide

```r
has_Muffin <- TRUE
shopping_at <- function (human = TRUE, location) {
 if (location == "bakery") {
* if (human == TRUE) {
 if (has_Muffin == TRUE) {
 bagel <- 10
 muffin <- 1
 } else {
 bagel <- 10
 }
* } else {
 if (has_Muffin == TRUE) {
 bagel <- 1
 } else {
 bagel <- 10
 }
 }
 }
 return(bagel)
}
shopping_at(human = TRUE, location = "bakery") # 10
shopping_at(human = FALSE, location = "bakery") # 1
```

---

## How about another fun slide?

To make things right, try to talk like this:

When you arrive at the bakery:

- If there are muffins, buy 1 muffin and 10 bagels. Checkout.

- If there is no muffin, buy 10 bagels. Checkout.

---

## So many fun slides

How we communicate:

- "Get some bagels. Maybe we should get some muffins. I don't know. If muffins look good, get some. Oh! Don't forget brownies. If you buy all three, get less bagels cause that will be too much. 
  
Human: totally fine! don't worry I got it!

Computer:

- error: "some" is not defined
  - logic error: "maybe"
  - logic error: "I don't know, either."
  - error: "good" is not defined
  - error: "less" is not defined
  - logic warning: "If you buy all three..."
  
---

## Today's topics

- Math Review
  - Ordinary Least Squares Linear Regression
- Programming in R
  - Simple regression analysis from scratch 
  
---
class: inverse, middle, center

# Math Review

---

## Linear Regression

There are many types of regression analysis.

In this course, we will focus on linear regression.

`$$Y_i = \beta_0 + \beta_1 \times X_i + \varepsilon$$`

where `$\beta_0$` is the intercept, `$\beta_1$` is the slope, and `$\varepsilon$` is the error.

---

## We need some data

Let's create 100 data points with two variables: `X` and `Y`, which are highly correlated.

$$
Y_i = X_i + \varepsilon
$$

```r
epsilon <- rnorm(n = 100, mean = 0, sd = 1)
dat_sim <- data.frame(
 X <- rnorm(n = 100, mean = 20, sd = 2),
 Y <- X + epsilon,
 diff = Y - X
)
colnames(dat_sim) <- c("X", "Y", "Diff.")
*head(dat_sim)
```

```
##          X        Y      Diff.
## 1 17.55815 18.06136  0.5032112
## 2 17.93834 17.03734 -0.9009984
## 3 22.20215 23.67810  1.4759490
## 4 19.12706 19.35034  0.2232747
## 5 22.18025 21.93926 -0.2409937
## 6 20.12013 20.38085  0.2607256
```

---

## Let's visualize the simulated data set

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```
## [1] "Correlation Coefficient: 0.874"
```

---

## We need a model

Let's fit a linear regression model. A linear regression defines a line:

`$$\hat{Y} = \beta_0 + \beta_1 X_i$$`

<div id="htmlwidget-2060c77765e9f70dc2ed" style="width:576px;height:432px;" class="plotly html-widget"></div>
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---

## What does this mean?

**Regression line** is the line that minimizes the **squared distances/errors** between `$Y_i$` and the line. In this case:

`$$(\beta_0, \beta_1) 
= \mathcal{argmin} \sum_{i=1}^{100} (Y_i - \hat {Y_i})^2
= \mathcal{argmin} \sum_{i=1}^{100} (Y_i - (\beta_0 + \beta_1 X_i))^2$$`

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---

## We love math!

`$$\begin{align}
\mathcal{Q}(\beta_0, \beta_1) & = \sum_{i=1}^{n} (Y_i - \hat {Y_i})^2 \\
& = \sum_{i=1}^{n} (Y_i - (\beta_0 + \beta_1 X_i))^2 \\
& = \sum_{i=1}^n (Y_i - \beta_0 -\beta_1 X_i)^2 \\
& = \sum_{i=1}^n Y_{i}^2 + n \beta_0^2 + \beta_1^2 \sum_{i=1}^n X_i^2 - 2 \beta_0 \sum_{i=1}^n Y_i - 2 \beta_1 \sum_{i=1}^n Y_i X_i + 2 \beta_0 \beta_1 \sum_{i=1}^n X_i \end{align}$$`

---

## We love math so much!

`$$\mathcal{Q}(\beta_0, \beta_1) = \sum_{i=1}^n Y_{i}^2 + n \beta_0^2 + \beta_1^2 \sum_{i=1}^n X_i^2 - 2 \beta_0 \sum_{i=1}^n Y_i - 2 \beta_1 \sum_{i=1}^n Y_i X_i + 2 \beta_0 \beta_1 \sum_{i=1}^n X_i$$`

`$$\frac{\partial Q}{\partial \beta_0} = 2n\beta_0 - 2n\overline{Y} + 2n\beta_1 \overline{X}$$`

`$$\frac{\partial Q}{\partial \beta_1} = 2\beta_1 \sum_{i=1}^n X_i^2 + 2n\beta_0 \overline{X} - 2n\sum_{i=1}^n Y_i X_i$$`

where `$\overline{Y}$` is the mean of `$Y$`, and `$\overline{X}$` is the mean of `$X$`.

---

## We love math! (Yes we do!)

`$\frac{\partial \mathcal{Q}}{\partial \beta_0}$` is the change of line equation if its intercept ($\beta_0$) is varied and its slope ($\beta_1$) is kept constant.

`$\frac{\partial \mathcal{Q}}{\partial \beta_1}$` is the change of line equation if its slope ($\beta_1$) is varied and its intercept ($\beta_0$) is kept constant.

---

## Almost there ...

To get the Ordinary **Least** Squares (OLS), we need to make sure there is no freedom of movement. This means, check if the second derivatives are positive.

`$$\frac{\partial \mathcal{Q}}{\partial \beta_0} = 
2n \beta_0 - 2n \overline{Y} + 2n \beta_1 \overline{X}$$`

`$$\frac{\partial \mathcal{Q}}{\partial \beta_0^2} = 2n$$`

`$$\frac{\partial \mathcal{Q}}{\partial \beta_1} = 2 \beta_1 \sum_{i=1}^n X_i^2 + 2n \beta_0 \overline{X} - 2n \sum_{i=1}^n Y_i X_i$$`

`$$\frac{\partial \mathcal{Q}}{\partial \beta_1^2} = 2 \sum_{i=1}^n X_i^2$$`

`$$\frac{\partial \mathcal{Q}}{\partial \beta_0 \beta_1} = 
2 \sum_{i=1}^n X_i = 2n \overline{X}$$`

---

## Final Math slide I promise

Then, solve `$\frac{\partial \mathcal{Q}}{\partial \beta_0}$` and `$\frac{\partial \mathcal{Q}}{\partial \beta_1}$`. We can get:

`$$\frac{\partial Q}{\partial \beta_0} = 2n\beta_0 - 2n\overline{Y} + 2n\beta_1 \overline{X}$$`

`$$\frac{\partial Q}{\partial \beta_1} = 2\beta_1 \sum_{i=1}^n X_i^2 + 2n\beta_0 \overline{X} - 2n\sum_{i=1}^n Y_i X_i$$`

`$$\beta_0 = \overline{Y} - \beta_1 \overline{X}$$`

`$$\beta_1 = \frac{\sum_{i=1}^n (X_i - \overline{X})(Y_i - \overline{Y})}
               {\sum_{i=1}^n (X_i - \overline{X})^2}
        = \frac{\sum_{i=1}^n (X_i - \overline{X})(Y_i - \overline{Y})}
               {n \times Var.(X_i)}
        = \frac{Corr.(X, Y) \times S.D.(Y)}{S.D.(X)}$$`

The higher the correlation, the higher the slope.

---

## Correlation and slope

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---
class: inverse, center, middle

# Programming in R

---

## Getting the data set

[2017 Cherry Blossom Run Data, 2017](https://www.openintro.org/data/index.php?data=run17)

[Download Here](https://www.openintro.org/data/csv/run17.csv)

Available on NYU Classes under the "Resources - Tong's Lab - Lab 3"

---

## The Data Set {.smaller}

Number of rows: 19,961

Variables

- bib: number on the runner's bib (ID)

- name: initials

- sex: biological sex

- age

- city: home city

- net_sec: time record in seconds (accounting for staggered starting time)

- clock_sec: time record in seconds (based on the clock)

- pace_sec: average time per mile, in seconds

- event: either the "10 Mile" race or the "5K"

---

## Import the data set to RStudio

```r
# Set working directory

# Load the data set
dat <- read.csv("run17.csv")
```

---

## Check dimensions

**Dimensions:** number of rows, number of columns

```r
# Count number of rows and save as N
*N <- nrow(dat)
```

```r
# Count number of columns
*ncol(dat)
# Check column types
*str(dat)
```

---

## Descriptive Analysis

```r
summary(dat)
```

```
##       bib                  name       sex            age       
##  Min.   :    3   Michael M.  :   30   F:12267   Min.   : 6.00  
##  1st Qu.: 6078   Jennifer S. :   28   M: 7694   1st Qu.:29.00  
##  Median :12128   John M.     :   26             Median :35.00  
##  Mean   :12300   Michael C.  :   25             Mean   :37.13  
##  3rd Qu.:18321   Stephanie S.:   24             3rd Qu.:44.00  
##  Max.   :25497   John S.     :   23             Max.   :92.00  
##                  (Other)     :19805             NA's   :1      
##                 city          net_sec       clock_sec       pace_sec     
##  Washington, DC   : 4861   Min.   :1078   Min.   :1078   Min.   : 279.0  
##  Arlington, VA    : 1789   1st Qu.:4767   1st Qu.:5048   1st Qu.: 524.0  
##  Alexandria, VA   :  921   Median :5658   Median :6333   Median : 596.0  
##  Silver Spring, MD:  534   Mean   :5428   Mean   :6083   Mean   : 608.7  
##  New York, NY     :  434   3rd Qu.:6433   3rd Qu.:7439   3rd Qu.: 674.0  
##  Bethesda, MD     :  390   Max.   :8398   Max.   :9896   Max.   :1759.0  
##  (Other)          :11032   NA's   :3      NA's   :3      NA's   :3       
##      event      
##  10 Mile:17442  
##  5K     : 2519  
##                 
##                 
##                 
##                 
## 
```

```r
# For the net_sec column
# mean(dat$net_sec)
# var(dat$net_sec)
# sd(dat$net_sec)
*sd(dat$net_sec) / sqrt(N)  # Standard error
```

```
## [1] NA
```

---

## Hypotheses

`$H_0$`: For 10 Mile race, the mean of the net finish time of male runners is the same as female runners.

`$H_1$`: For 10 Mile race, the mean of the net finish time of male runners is **not** the same as female runners.

---

## Subset the data set

```r
dat_10m <- dat[dat$even == "10 Mile", ]
dat_10m$sex <- ifelse(dat_10m$sex == "M", 1, 0)
```

---

## Levene's test

To examine if pooled variance is suitable.

```r
leveneTest(net_sec ~ factor(sex), data = dat_10m)
```

```
## Levene's Test for Homogeneity of Variance (center = median)
##          Df F value    Pr(>F)    
*## group     1  34.267 4.891e-09 ***
##       17440                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```

---

## T-test

```r
t.test(net_sec ~ sex, data = dat_10m, var.equal = FALSE)
```

```
## 
## 	Welch Two Sample t-test
## 
## data: net_sec by sex
*## t = 40.434, df = 14287, p-value < 2.2e-16
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## 582.1310 641.4466
## sample estimates:
## mean in group 0 mean in group 1 
## 6113.618 5501.829
```

---

## Linear regression

```r
summary(lm(net_sec ~ sex, data = dat_10m))
```

```
## 
## Call:
## lm(formula = net_sec ~ sex, data = dat_10m)
## 
## Residuals:
## Min 1Q Median 3Q Max 
## -2896.62 -687.62 -38.62 640.33 2896.17 
## 
## Coefficients:
## Estimate Std. Error t value Pr(>|t|) 
## (Intercept) 6113.618 9.454 646.69 <2e-16 ***
## sex -611.789 14.927 -40.98 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 966.2 on 17440 degrees of freedom
## Multiple R-squared: 0.08786,	Adjusted R-squared: 0.0878 
## F-statistic: 1680 on 1 and 17440 DF, p-value: < 2.2e-16
```

---

## Visualize regression results

```r
*plot(x = dat_10m$sex, y = dat_10m$net_sec, type = "p")
abline(a = 6113.618, b = -611.789, lwd = 2)
```

![](data:image/png;base64,#Lab3-Slides-092220_files/figure-html/viz_lm1-1.png)

---

## Let's do another regression analysis

**On average, the older, the slower.**

```r
summary(lm2 <- lm(net_sec ~ age, data = dat_10m))
```

```
## 
## Call:
## lm(formula = net_sec ~ age, data = dat_10m)
## 
## Residuals:
## Min 1Q Median 3Q Max 
## -2969.38 -700.70 -20.42 672.78 2670.51 
## 
## Coefficients:
## Estimate Std. Error t value Pr(>|t|) 
## (Intercept) 5388.0324 27.2989 197.37 <2e-16 ***
## age 12.9782 0.7087 18.31 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1002 on 17440 degrees of freedom
## Multiple R-squared: 0.01886,	Adjusted R-squared: 0.01881 
## F-statistic: 335.3 on 1 and 17440 DF, p-value: < 2.2e-16
```

---

## Visualize regression results

```r
*plot(x = dat_10m$age, y = dat_10m$net_sec, type = "p", col = "grey")
abline(a = 5388.0324, b = 12.9782, lwd = 2)
```

![](data:image/png;base64,#Lab3-Slides-092220_files/figure-html/viz_lm2-1.png)

---

## De-mean

```r
*dat_10m$age0 <- dat_10m$age - mean(dat_10m$age)
summary(lm3 <- lm(net_sec ~ age0, data = dat_10m))
```

```
## 
## Call:
## lm(formula = net_sec ~ age0, data = dat_10m)
## 
## Residuals:
## Min 1Q Median 3Q Max 
## -2969.38 -700.70 -20.42 672.78 2670.51 
## 
## Coefficients:
## Estimate Std. Error t value Pr(>|t|) 
## (Intercept) 5868.2292 7.5877 773.38 <2e-16 ***
## age0 12.9782 0.7087 18.31 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1002 on 17440 degrees of freedom
## Multiple R-squared: 0.01886,	Adjusted R-squared: 0.01881 
## F-statistic: 335.3 on 1 and 17440 DF, p-value: < 2.2e-16
```

---

## Visualize regression results

```r
plot(x = dat_10m$age0, y = dat_10m$net_sec, type = "p", col = "grey")
abline(a = 5868.2292, b = 12.9782, lwd = 2)
abline(v = 0, lty = "dashed", col = "red")
```

![](data:image/png;base64,#Lab3-Slides-092220_files/figure-html/viz_lm3-1.png)

---

## Standardize

```r
dat_10m$age1 <- dat_10m$age0 / sd(dat_10m$age0)
summary(lm4 <- lm(net_sec ~ age1, data = dat_10m))
```

```
## 
## Call:
## lm(formula = net_sec ~ age1, data = dat_10m)
## 
## Residuals:
## Min 1Q Median 3Q Max 
## -2969.38 -700.70 -20.42 672.78 2670.51 
## 
## Coefficients:
## Estimate Std. Error t value Pr(>|t|) 
## (Intercept) 5868.229 7.588 773.38 <2e-16 ***
## age1 138.950 7.588 18.31 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1002 on 17440 degrees of freedom
## Multiple R-squared: 0.01886,	Adjusted R-squared: 0.01881 
## F-statistic: 335.3 on 1 and 17440 DF, p-value: < 2.2e-16
```

---

## Visualize regression results

```r
plot(x = dat_10m$age1, y = dat_10m$net_sec, type = "p", col = "grey", xlab = "SD")
abline(a = 5868.229, b = 138.950, lwd = 2)
abline(v = 0, lty = "dashed", col = "red")
```

![](data:image/png;base64,#Lab3-Slides-092220_files/figure-html/viz_lm4-1.png)